Density patterns in two-dimensional hoppers

Recent experiments by Baxter et al. [Phys. Rev. Lett. 62, 2825 (1989)] showed the existence of density waves in granular material flowing out of a hopper. We show, using molecular dynamics simulations, that this effect is a consequence of static friction and find that these density fluctuations follow a 1/${\mathit{f}}^{\mathrm{\alpha }}$ spectrum. The effect is enhanced when the opening angle of the hopper decreases.